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Flows in Complex Networks: Theory, Algorithms, and Application to Lennard–Jones Cluster Rearrangement

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Abstract

A set of analytical and computational tools based on transition path theory (TPT) is proposed to analyze flows in complex networks. Specifically, TPT is used to study the statistical properties of the reactive trajectories by which transitions occur between specific groups of nodes on the network. Sampling tools are built upon the outputs of TPT that allow to generate these reactive trajectories directly, or even transition paths that travel from one group of nodes to the other without making any detour and carry the same probability current as the reactive trajectories. These objects permit to characterize the mechanism of the transitions, for example by quantifying the width of the tubes by which these transitions occur, the location and distribution of their dynamical bottlenecks, etc. These tools are applied to a network modeling the dynamics of the Lennard–Jones cluster with 38 atoms (\(\mathrm{LJ}_{38}\)) and used to understand the mechanism by which this cluster rearranges itself between its two most likely states at various temperatures.

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Notes

  1. Note that in [24], (9) was referred to as the current of reactive trajectories and (10) as the effective current of reactive trajectories: the terminology used here is more consistent with standard conventions in which a current should be antisymmetric in its indices.

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Acknowledgments

We thank Prof. David Wales for providing us with the data of the \(\mathrm{LJ}_{38}\) network and Miranda Holmes-Cerfon for interesting discussions. M. C. held an Sloan Research Fellowship and was supported in part by DARPA YFA Grant N66001-12-1-4220, and NSF Grant 1217118. E. V.-E. was supported in part by NSF Grant DMS07-08140 and ONR Grant N00014-11-1-0345.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Maria Cameron

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Eric Vanden-Eijnden

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  1. Maria Cameron
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  2. Eric Vanden-Eijnden
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Correspondence to Eric Vanden-Eijnden.

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Cameron, M., Vanden-Eijnden, E. Flows in Complex Networks: Theory, Algorithms, and Application to Lennard–Jones Cluster Rearrangement. J Stat Phys 156, 427–454 (2014). https://doi.org/10.1007/s10955-014-0997-8

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  • DOI: https://doi.org/10.1007/s10955-014-0997-8

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